If the original motor mounts are still in the car, they're probably broken or close if >10 years old or so.
All three mounts stabilize the motor and reduce its movement within the engine bay, but I believe having a firmer rear motor mount makes the biggest difference (if the other two are decent)
Most of the effect will be felt if drive a manual because if the engine is held solid it'll be less prone to lurching.
"testing" is limited to trying to move in 1st/R while holding the brake and seeing how much the engine rotates in the bay. Other than that, just lifting the load off and checking to see if it's broken/separated